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Finding a pdf from a cdf of a CRV

Finding a pdf from a cdf of a CRV

IB Year 5 | Grade 11

\(2.\) The continuous random variable \(X\) has probability density function given by

\[\mathrm{f}(x) = \begin{cases} k(2 - x), & \text{for}\hspace{0.5em} 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases}, \text{ where}\hspace{0.5em} k \text{ is a constant.}\]

  1. Find the value of \(k\).
  2. Find the cumulative distribution function of \(X\).
  3. The continuous random variable \(Y\) is given by \(Y = 1 - \frac{1}{2}X\).
    1. Show that \(\mathrm{P}(Y < y) = y^2\), where \(0 \le y \le 1\).
    2. Deduce the probability density function of \(Y\) and hence, or otherwise, show that \[E(Y) = \frac{2}{3}.\]

Solution:

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Answer:(a) \(k=\frac12\). (b) \(\mathrm{F}_X(x)=\begin{cases}0,&x<0,\\ x-\frac{x^2}{4},&0\leq x\leq2,\\ 1,&x>2.\end{cases}\) (c)(i) \(\mathrm{P}(Y<y)=y^2\) for \(0\leq y\leq1\). (ii) \(\mathrm{f}_Y(y)=2y\) for \(0<y<1\), zero otherwise; \(\mathrm{E}(Y)=\frac23\).

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